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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A comparison theorem for linear difference equations


Author: P. W. Eloe
Journal: Proc. Amer. Math. Soc. 103 (1988), 451-457
MSC: Primary 39A10
DOI: https://doi.org/10.1090/S0002-9939-1988-0943065-5
MathSciNet review: 943065
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Abstract: Let a be real, $ I = \{ a,a + 1, \cdots ,b\} $ where $ b - a$ is a positive integer or $ I = \{ a,a + 1, \ldots \} $. Let $ n$ be a positive integer and let $ {I^n} = \{ a,a + 1, \ldots ,b + n\} $ if $ b < \infty \;{\text{or}}\;{I^n} = I$ otherwise. Consider the $ n$th order difference equation $ Pu(m) = \sum\nolimits_{j = 0}^n {{\alpha _j}(m)u(m + j) = 0,\;{\alpha _n}(m) = 1,\;{\alpha _0}(m) \ne 0,\;m \in I} $. It is shown that if $ 0 \leq r(m) \leq q(m),\;m \in I$ and if the equations $ Pu(m) = 0$ and $ Pu(m) + q(m)u(m) = 0$ are disconjugate on $ {I^n}$, then the equation $ Pu(m) + r(m)u(m) = 0$ is disconjugate on $ {I^n}$.


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DOI: https://doi.org/10.1090/S0002-9939-1988-0943065-5
Keywords: Disconjugacy, nonoscillation, comparison theorem, linear difference equation
Article copyright: © Copyright 1988 American Mathematical Society